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A second order SDE for the Langevin process reflected at a completely inelastic boundary
Jean Bertoin
To cite this version:
Jean Bertoin. A second order SDE for the Langevin process reflected at a completely inelastic bound- ary. 2006. �hal-00106365�
ccsd-00106365, version 1 - 14 Oct 2006
A second order SDE
for the Langevin process reflected at a completely inelastic boundary
Jean Bertoin
Laboratoire de Probabilit´es et Mod`eles Al´eatoires Universit´e Pierre et Marie Curie
and DMA, Ecole Normale Sup´erieure Paris, France
Summary. It was shown in [2] that a Langevin process can be reflected at an energy absorbing boundary. Here, we establish that the law of this reflecting process can be characterized as the unique weak solution to a certain second order stochastic differential equation with constraints, which is in sharp contrast with a deterministic analog.
Key words. Langevin process, reflection, stochastic differential equation.
A.M.S. Classification. Primary 60 H 10 , 60 J 55. Secondary 34 A 12 e-mail. [email protected]
1 Introduction
Consider the motion of a particle in the half-line R+ under an external force that governs its acceleration. Assume that the energy of the particle is instantaneously absorbed at the boundary point 0, meaning that the velocity of the particle is always 0 immediately after hitting 0. In other words, the trajectory (Xt)t≥0 of the particle fulfills the constraints of completely inelastic impacts
Xt≥0,
Xt= 0 ⇒ X˙t+ = 0. (1)
and solves the second order differential equation
dXt = ˙Xtdt , d ˙Xt=Ftdt+ dAt , At=− X
0<s≤t
X˙s−1{Xs=0}, (2) where (Ft)t≥0 denotes the external force. More precisely, t → At is right-continuous non- decreasing and accounts for the kick induced by the boundary. Specifically, if the particle
hits 0 at time t with incoming velocity ˙Xt− < 0, then At−At− = −X˙t− > 0 so that ˙Xt+ = X˙t−+ (At−At−) = 0.
Equation (2) can be viewed as a special case of differential measure inclusions which have been studied initially by Schatzman [13]; see also Ballard [1] and the references therein. It is quite remarkable that multiple solutions may exist even in situations when the external force isC∞.
Following a question raised by Bertrand Maury, we are interested in the case when the external force is a generalized function given by a white noise, i.e. when Ft = ˙Bt with (Bt)t≥0
a standard Brownian motion. In this setting, it is natural to consider first the much simpler situation when there is no obstacle at 0, that is to introduce the process with values in R
Yt =y0+ty˙0+ Z t
0
Bsds , t ≥0.
The latter will be called here a free Langevin process, started from location y0 ∈ R and with initial velocity ˙y0 ∈ R; we refer to Lachal [5] for a rich source of results and references in this area. It is easily seen that for y0 = ˙y0 = 0, 0 is an accumulation point of the set of times at which the free Langevin process returns to 0. Informally, this may suggest that if the energy of the Langevin particle is absorbed at each visit to 0, then the particle might never be able to reach a strictly positive velocity, and thus might never take off the boundary. It turns out that this intuition is not correct as we shall see.
It is convenient to agree that throughout this work, all random processes are implicitly c`adl`ag, i.e. their sample paths are right-continuous and possess left-limits everywhere, a.s.
In a preceding work [2], we established the following result of existence and uniqueness in distribution.
Theorem 1 There exists a strong Markov process (Xt,X˙t)t≥0 with values in R+×R, started from X0 = ˙X0 = 0, such that
dXt= ˙Xtdt , Z ∞
0
1{Xt=0}dt = 0 and Xt= 0 ⇒ X˙t= 0 a.s., (3) and which evolves as a free Langevin process as long as X >0. Specifically, for every stopping time S in the natural filtration of X (after the usual completions), if we define ζS = inf{t ≥ S : Xt = 0}, then conditionally on XS = x0 > 0 and X˙S = ˙x0, the process (XS+t)0≤t≤ζS is independent of FS and has the same distribution as (Yt)0≤t≤ζ, where
Yt=x0+tx˙0+ Z t
0
Bsds , ζ = inf{t≥0 :Yt= 0}
and(Bt)t≥0 is a standard Brownian motion. Further, the preceding requirements determine the distribution of (Xt,X˙t)t≥0.
We stress that the strong Markov process (Xt,X˙t)t≥0 has jumps at predictable stopping times (namely, the hitting times by X of the boundary point 0), and thus fails to be standard; in particular, the Feller property does not hold.
The main purpose of this work is to connect the process characterized in Theorem 1 to Equation (2) when the external force F is a white noise. In this direction, it is convenient to rewrite (2) in the form
dXt= ˙Xtdt , X˙t =Bt+At , At =− X
0<s≤t
X˙s−1{Xs=0}. (4)
We are now able to state the main result of this work.
Theorem 2 (i)One can construct on some filtered probability space (Ω,(Ft)t≥0,P)an adapted process (Xt)t≥0 distributed as in Theorem 1 and an (Ft)-Brownian motion (Bt)t≥0, such that Equations (1) and (4)hold.
(ii) Conversely, if on some filtered probability space (Ω,(Ft)t≥0,P), there is an (Ft)-Brownian motion (Bt)t≥0 and an adapted process(Xt)t≥0 which satisfies Equations (1)and (4)and starts with initial conditions X0 = ˙X0 = 0, then (Xt)t≥0 is distributed as in Theorem 1.
We shall refer to the process X which appears in Theorems 1 and 2 as a Langevin process reflected at a completely inelastic boundary. Note that we implicitly restrict our attention to the case when the process starts from 0 with initial velocity 0, which is obviously the most interesting situation and induces no loss of generality. In some loose sense, Theorems 1 and 2 both state the existence and uniqueness in law of the Langevin process reflected at completely inelastic boundary, but viewed from two different perspectives. Theorem 1 belongs to the framework of the theory of Markov processes and their excursions, whereas Theorem 2 is expressed in terms of stochastic differential equations. It is well-known that these two theories are intimately connected, and one may expect that a soft argument should enable us to deduce Theorem 2 from Theorem 1.
In this direction, the existence of a weak solution to (4) and (1) is rather easy and will be established in the first part of Section 2 by investigating, in the framework of stochastic calculus, the explicit construction given in [2] of the process specified by Theorem 1. More precisely, the latter is obtained from the free Langevin process associated to some standard Brownian motion (Wt)t≥0 first by a reflection `a la Skorohod and then by a non-invertible random time-substitution.
However, establishing weak uniqueness in Theorem 2 is less straightforward. Indeed, if we aim at applying Theorem 1, then we have to checka priori that any weak solution (Xt,X˙t)t≥0
to (4) and (1) enjoys the strong Markov property. But it is well-known that solutions of an SDE fulfill the Markov property only in the situation when weak uniqueness holds for the SDE, and thus Theorem 1 cannot help. We also stress that weak uniqueness is the most striking aspect of Theorem 2 as it is in sharp contrast with the deterministic situation for which (strong) uniqueness can fail even with a smooth forcing.
In the second part of Section 2, we shall observe a key point which lies at the heart of the proof of weak uniqueness. From the same Brownian motion (Wt)t≥0 which is used to construct a weak solution (Xt, Bt)t≥0, one can also built another standard Brownian motion (Bt′)t≥0 which is independent of (Bt)t≥0, and such that (Wt)t≥0 can be recovered from (Xt, Bt)t≥0 and (Bt′)t≥0.
Weak uniqueness is established in Section 3. We consider any solution (Xt, Bt)t≥0 to (4) and (1) where (Bt)t≥0 is some Brownian motion. We then introduce an independent standard Brownian motion (Bt′)t≥0, and using the analysis developed in Section 2, we construct from (Xt, Bt)t≥0
and (Bt′)t≥0 another Brownian motion (Wt)t≥0, such that (Xt)t≥0 can be recovered from (Wt)t≥0
in the same way as in Section 2.
In the final Section, we first make some brief historical comments about the question of uniqueness in the -deterministic- setting of mechanical systems with perfect constraints. For the reader’s convenience, we also provide a simple example showing that uniqueness of the solution to (1) and (2) may fail even when the external force is smooth. Finally, we discuss some open questions regarding strong solutions to (4) and (1).
Nota Bene. In this paper, I will use the same notationX, B, W,... for processes which,in fine, will be shown to have the same distributions. However the initial definition and assumptions for these processes may be different in different sections. I hope that the reader will find this helpful and not confusing.
2 A weak solution
The first purpose of this section is to check that the construction of Section 2 in [2] also provides a solution to (4) and (1). Then we shall study this construction in further details to gain insight for the proof of weak uniqueness.
2.1 Construction of a weak solution
We start by recalling the construction of Section 2 in [2] and some of its properties.
Let W = (Wt, t≥ 0) be a standard Wiener process started from W0 = 0 and write (Wt)t≥0
for its natural filtration after the usual completions. Define the free Langevin process Yt :=
Z t 0
Wsds , t≥0, its infimum process
It:= inf{Ys: 0≤s≤t},
and the random closed set of times when Y coincides with its infimum I :={t≥0 :Yt=It}.
We write Io for the interior of I and recall from Lemma 2 in [2] that with probability one, the boundary ∂I =I\Io has zero Lebesgue measure. Further the canonical decomposition of the open set Io into disjoint open intervals is given by
Io = [
u∈J
]u, du[,
where J is the set of times at which Y reaches its infimum for the first time during some negative excursion of W, and du the first return time to 0 for W after the instant u. That is
J :={t ≥0 :Wt<0, Yt=It and Yt−ε > It−ε for all ε >0 sufficiently small}
and
du := inf{s > u:Ws = 0}.
It is readily seen that J can be expressed in the form of a countable family of stopping times in the filtration (Wt)t≥0. For instance J ={Sk,n : k, n ∈ N}, where Sk,n is the k-th instant t such that Yt =It, Yt−ε > It−ε for all ε > 0 sufficiently small and the velocity at timet fulfills Y˙t∈]−1/(n−1),−1/n]. Note that the dSk,n are then also stopping times.
Finally we introduce the right-continuous time-substitution Tt:= inf
s≥0 : Z s
0
1{Yv>Iv}dv > t
, t≥0,
and then the free Langevin process reflected at its infimum (in the sense of Skorohod) and time-changed by Tt, that is we set for everyt ≥0
Xt:= (Y −I)◦Tt.
We mention that the process t → Yt−It has been studied first by Lapeyre [6]. Clearly, the processt→Xtonly takes nonnegative values, is continuous, and it can be shown that it possess a right-derivative at every t≥0 given by
X˙t =W ◦Tt; see Equation (7) in [2]. We also set Ft=WTt.
The following proposition establishes the existence stated in Theorem 2(i).
Proposition 1 Define At:=
Z Tt
0
1{Ys=Is}dWs and Bt:=
Z Tt
0
1{Ys>Is}dWs, t≥0, so that
X˙t=At+Bt, t≥ 0. Then there is the identity
At =− X
0<s≤t
X˙s−1{Xs=0}, t≥0,
and (Bt)t≥0 is an (Ft)-Brownian motion. As a consequence, (Xt,X˙t)t≥0 is a weak solution to (4)and (1) with initial condition X0 = ˙X0 = 0.
Remark : The fact that the series P
0<s≤tX˙s−1{Xs=0} converges for every t ≥0 a.s. can be deduced from Corollary 2 in [2]. However this fact shall be established directly in the present analysis.
Proof: We express ˙Xt = W ◦Tt = At+Bt, where At and Bt are defined in the statement.
The basic facts that have been recalled above imply the identities Z t
0
1{Ys=Is}dWs= Z t
0
1{s∈I}dWs = Z t
0
1{s∈Io}dWs =X
u∈J
(Wdu∧t−Wu∧t). (5)
On the one hand, the assertion that (Bt)t≥0 is an (Ft)-Brownian motion is seen from the very definition of the time-substitution Tt and the Dambis-Dubins-Schwarz theorem (see e.g.
[12] on its page 181). On the other hand, again by definition, Tt 6∈ Io and Wdu = 0 for every u∈ J. We deduce from (5) the identity
At=− X
u∈J,u≤Tt
Wu.
Further, it is easily checked that J coincides with the set of times of the form u = Ts− with s >0 an instant at which X hits the boundary point 0 with a negative incoming velocity (i.e.
Xs = 0 andXs−<0). Since ˙Xs−=W ◦Ts−, we conclude that At=− X
0<s≤t
X˙s−1{Xs=0}, t≥0.
To complete the proof, either we observe that if t is an instant at which Xt = 0, then X˙t− =Bt+At− and thus
X˙t =Bt+At= ˙Xt−+ (At−At−) = ˙Xt−−X˙t−= 0,
or we just recall from Equation (3) in Theorem 1 that Xt= 0⇒X˙t = 0.
2.2 Some further properties
We introduce the time-substitution Tt′ := inf
s≥0 : Z s
0
1{Yv=Iv}dv > t
, t ≥0, which can be thought of as the dual toTt. Next we set
Bt′ :=
Z Tt′ 0
1{Yv=Iv}dWv, t ≥0, and then, for every x≥0,
σ′(x) := inf{t ≥0 :Bt′ > x}
for the first passage time of B′ above the level x.
Lemma 1 With probability one, there is the identity
Tt=t+σ′(At), t≥0.
Proof: It will be convenient to write B′(t) := Bt′ and observe from the definition of B′ and At (in Proposition 1) the identities
σ′(At) = inf Z s
0
1{Yv=Iv}dv :B′ Z s
0
1{Yv=Iv}dv
> At
= inf Z s
0
1{Yv=Iv}dv : Z s
0
1{Yv=Iv}dWv >
Z Tt
0
1{Yv=Iv}dWv
.
Then recall (5). Observe that for everyu∈ J, the processs →Wdu∧s−Wu∧sis 0 before time u, takes some strictly positive values immediately after time u, reaches its overall maximum for the first time at du, and remains constant after du. Note furthermore that the intervals [u, du] foru∈ J are pairewise disjoint. It follows that whenever t 6∈ Io, the stochastic integral s→Rs
0 1{Yv=Iv}dWv attains its overall maximum on the time-interval [0, t] at time t, and if we definer(t) = inf{s > t :s∈ Io}, then the first instant when this stochastic integral exceeds its value at time t isr(t). Further this stochastic integral remains constant on [t, r(t)].
Applying these observations to the random time Tt6∈ Io, we conclude that σ′(At) =
Z r(Tt) 0
1{Yv=Iv}dv = Z Tt
0
1{Yv=Iv}dv , and thus
Tt = Z Tt
0
1{Yv>Iv}dv+ Z Tt
0
1{Yv=Iv}dv =t+σ′(At),
as it has been stated.
We are now able to establish the following statement, which will provides us with the hint for establishing weak uniqueness in the next section.
Proposition 2 The process (B′t)t≥0 is a standard Brownian which is independent of (Bt)t≥0. Further, W can be recovered from (X, B, B′) as
Wt=Bτ(t)+Bτ′′(t), where
τ(t) := inf{s≥0 :s+σ′(As)> t} and τ′(t) := 1−τ(t).
Proof: That (B′t)t≥0 is a Brownian motion which is independent of (Bt)t≥0 follows immediately from the definition ofB andB′ and Knight’s extension of the Dambis-Dubins-Schwarz theorem (see e.g. [12] on its page 183).
Then we simply write Wt =
Z t 0
1{Ys>Is}dWs+ Z t
0
1{Ys=Is}dWs =Bτ(t)+Bτ′′(t), where
τ(t) :=
Z t 0
1{Ys>Is}ds and τ′(t) :=
Z t 0
1{Ys=Is}ds .
The identity τ(t) +τ′(t) = t is obvious. By definition, Tt = inf{s ≥ 0 : τ(s) > t} and thus τ(·) coincides with the continuous left-inverse of the strictly increasing time-change T·, i.e.
τ(Tt)≡t. We get from Lemma 1 that τ(t) = inf{s ≥0 :s+σ′(As)> t}.
Remark : By a more careful analysis, one could also establish a stronger result of inde- pendence, namely that X and B′ are independent processes. Nonetheless, as this will not be needed in this work and also follows from the analysis in the next section, we leave the direct proof to the interested reader. In this direction, we also stress that the Brownian motion B is adapted to the natural filtration of X, as one sees from Proposition 1. But we do not know whether, conversely, X is adapted to the natural filtration of the Brownian motion B, that is whether the solution to (4) is strong.
3 Uniqueness in distribution
In this Section, we consider some filtered probability space (Ω,(Ft)t≥0,P). We assume there is an (Ft)-Brownian motion (Bt)t≥0 and an adapted process (Xt)t≥0 which satisfies Equations (1) and (4) and starts with initial conditions X0 = ˙X0 = 0. Our goal is to establish that (Xt)t≥0 has the distribution of the process in the preceding Section. In this direction, Proposition 2 points at the role of an independent Brownian motion, so we assume that the same probability space Ω can be endowed with another filtration (Ft′)t≥0 such that the terminal sigma-fieldsF∞ andF∞′ are independent, and that there exists an (Ft′)-Brownian motion (Bt′)t≥0. Clearly, these assumptions induce no loss of generality (as it suffices to enlarge the initial probability space).
Just as in the preceding Section, we then write
σ′(x) := inf{t ≥0 :Bt′ > x}
for the first passage time of B′ above level x≥0, and define Tt :=t+σ′(At), t ≥0.
The process t→Tt is strictly increasing and thus possesses a continuous left-inverse τ(t) := inf{s ≥0 :s+σ′(As)> t}, t ≥0,
i.e. τ(Tt)≡t. Clearly 0 ≤τ(t)≤t, and we also set
τ′(t) :=t−τ(t), t≥0.
Finally we define
Wt:=Bτ(t)+Bτ′′(t), t ≥0.
The weak uniqueness stated in Theorem 2(ii) is now a consequence of the following.
Proposition 3 (i)The process (Wt)t≥0 is a standard Brownian motion.
(ii) The process (Xt)t≥0 can be recovered as
Xt = (Y −I)◦Tt, t≥0, where
Yt :=
Z t 0
Wsds and It:= inf{Ys : 0≤ s≤t}. (iii) Finally, there is the identity
Tt= inf
s≥0 : Z s
0
1{Yv>Iv}dv > t
, t≥0.
Remark : Proposition 3 shows thatX is distributed as the process which appears in Theorem 1, and as a consequence, we must haveR∞
0 1{Xt=0}dt= 0 a.s. It may be interesting to point out that this property can be checked directly from (4) and (1). More precisely, the set of times t whenXt= 0 is contained into the zero set of the Brownian semi-martingale ˙X =B+A. That the latter has zero Lebesgue measure a.s. can be seen from the occupation density formula for Brownian semi-martingales, see e.g. Corollary 1 in [11] on its page 216.
The rest of this Section is devoted to the proof of Proposition 3; we start with the first part.
Proof of (i): Although one may perhaps establish the result more directly by stochastic calculus, we shall use an approximation, as this makes the proof more intuitive. Specifically, pick ε >0 and introduce
A(ε)t :=− X
0<s≤t
X˙s−1{Xs=0,X˙s−<−ε}, t≥0,
so thatt →A(ε)t is a non-decreasing process and
limε↓0 ↑ A(ε)t =At, t ≥0. Set also
τ(ε)(t) := inf{s ≥0 :s+σ′(A(ε)s )> t},
sot →τ(ε)(t) is a continuous non-decreasing process with 0≤τ(ε)(t)≤t and limε↓0 ↓τ(ε)(t) =τ(t), t≥0.
Thus, if we define
Wt(ε) :=Bτ(ε)(t)+B′t−τ(ε)(t), t≥0,
then we have
ε→0+lim Wt(ε)=Wt. (6)
Hence, it now suffices to check that for every ε > 0, the process (Wt(ε))t≥0 is a standard Brownian motion. Let us first provide an intuitive explanation. The time-change t→τ(ε)(t) is an absolutely continuous process, and its derivative dτ(ε)(t)/dt:= ˙τ(ε)(t) is a step process which takes alternately the values 1 and 0. The dual time-change t → t−τ(ε)(t) is also absolutely continuous with derivative 1−τ˙(ε)(t)∈ {0,1}. The process W(ε) is thus obtained by following alternately the paths of two independent Brownian motions, B and B′, in a way which may remind us of the classical two-arm bandit (see, e.g. [4]). More precisely, W(ε) follows B when
˙
τ(ε)(t) = 1 and followsB′ otherwise. The instants whenW(ε)switches fromB toB′ correspond to the jump times of A(ε), whereas the instants when W(ε) switches from B′ to B correspond to certain first passage times of B′. These switching times form an increasing sequence of predictable random times for W(ε), and we can then deduce from the strong Markov property that W(ε) is a standard Brownian motion.
More precisely, the assumption that X solves (4) and elementary properties of the free Langevin process easily imply that with probability 1, the set of times t at which Xt hits the boundary 0 with incoming velocity ˙Xt− <−ε is both discrete and unbounded. Thus the set of jump times ofA(ε)can be expressed as an increasing sequence of stopping timesJ1(ε) < J2(ε) <· · · where J0(ε)= 0 and
Jn(ε) := inf{t > Jn−1(ε) :Xt= 0 and ˙Xt− <−ε}, n∈N, and limn→∞Jn(ε) =∞.
Write for simplicitya(ε)n =A(ε)
Jn(ε)
, and consider the increasing sequence (σ′(a(ε)n ))n∈N. As each a(ε)n is a random variable which is measurable with respect to F∞ and thus independent of B′, theσ′(a(ε)n ) form an increasing sequence of randomized (Ft′)-stopping times. The strong Markov property entails that conditionally on (a(ε)n )n∈N, the pieces of Brownian paths
(B′
t+σ′(a(ε)n−1)−a(ε)n−1 : 0≤t < σ′(a(ε)n )−σ′(a(ε)n−1))
are independent, and for each fixed n ∈ N, the conditional law of this n-th piece is that of a standard Brownian motion killed when it exceeds a(ε)n −a(ε)n−1.
The process (Wt(ε))t≥0 is obtained by splicing the sequence of pieces of paths (Bt: 0≤t < J1(ε)),(Bt′ : 0≤t < σ′(a(ε)1 )),(Bt+J(ε)
1 −BJ(ε)
1 : 0≤t < J2(ε)−J1(ε)), . . . In particular
Wt(ε)=
( Bt when 0≤t < J1(ε), B′
t−J1(ε)+BJ(ε)
1 when J1(ε) ≤t < J1(ε)+σ′(a(ε)1 ),
and the strong Markov property of Brownian motion shows that the process (Wt(ε) : 0 ≤ t <
J1(ε)+σ′(a(ε)1 )) has the same law as (Bt: 0≤t < ρ1) where ρ1 is the (Ft)-stopping time defined
as ρ1 := inf{t > J1(ε) :Bt−BJ(ε)
1 > a(ε)1 }. Splicing more and more pieces, we now see by that an iteration of this argument that (Wt(ε) : 0≤t < Jn(ε)+σ′(a(ε)n )) has the same distribution as (Bt : 0 ≤ t < ρn) where ρn := inf{t > Jn(ε) : Bt−BJ(ε)
n > a(ε)n }. Letting n → ∞, we conclude that (Wt(ε))t≥0 is a standard Brownian motion, and an appeal to (6) completes the proof.
Next, let us write
DA:={t >0 :At> At−}
for the set of times whereA is discontinuous. Observe from (4) that we have also the identifi- cation
DA={t >0 :Xt= 0 and ˙Xt− <0}
as the set of instants when X hits the boundary 0 with a strictly negative incoming velocity.
An important step in the proof of Proposition 3 is provided by the following representation of the processes τ(·) and τ′(·).
Lemma 2 Introduce the random open set O := [
t∈DA
]t+σ′(At−), t+σ′(At)[, and write Oc := [0,∞[\O for its complementary set.
(i)The processest →τ(t)andt→τ′(t)are both absolutely continuous non-decreasing processes with Stieltjes measures given by
dτ(t) =1Ocdt , dτ′(t) =1Odt . (ii) We have Wt≤ 0and Xτ(t)= 0 for every t ∈ O, a.s.
Proof: (i) Write λ for the Lebesgue measure onR+. We have to show that
λ(O ∩[0, t]) = τ′(t), t≥0. (7)
In this direction, let A denote the closed range of the process A·, viz. the set of points of the type At or At− for some t ≥ 0. The complementary set Ac := R+\A has a canonical decomposition as union of disjoint open intervals given by
Ac = [
t∈DA
]At−, At[.
Observe that, since A· is a pure jump process, then for every t ≥0 λ([0, At]\A) = X
s∈DA∩[0,t]
(As−As−) =At, and hence λ(A) = 0.
On the other hand, it is well-known that the first passage processσ′ is a stable subordinator with index 1/2. In particular, it is purely discontinuous, and, by the L´evy-Itˆo decomposition, the process of its jumps is a Poisson point process. Since A has zero Lebesgue measure and is independent ofσ′,A does not contain any jump time of σ′, a.s. It follows that for every v ≥0
λ(O ∩[0, v+σ′(Av)]) = X
s∈DA∩[0,v]
(σ′(As)−σ′(As−)) = σ′(Av). Recall now thatTv =v+σ′(Av) and observe that
τ′(Tv) =Tv−τ(Tv) = Tv−v =v+σ′(Av)−v =σ′(Av).
We have thus checked that (7) holds for every t of the form t =Tv for some v ≥0, and hence, by approximation, also for everyt of the formt =Tv− for somev >0.
Finally, suppose that t ∈]Tv−, Tv[, where v =τ(t). We get from above λ(O ∩[0, t]) = λ(O ∩[0, Tv−]) +t−Tv− =τ′(Tv−) +t−Tv−. But
τ′(Tv−) +t−Tv− =Tv−−τ(Tv−) +t−Tv− =t−v =t−τ(t) =τ′(t), and we conclude that (7) holds for allt ≥0.
(ii) If t∈ O, then t∈]Tv−, Tv[ where v =τ(t). By definition, we have Wt =Bv+Bt−v′ .
On the other hand,v ∈ DAand thusXv = 0. Further, by (4) and (1), we have ˙Xv =Bv+Av = 0.
We deduce that
Wt=Bt−v′ −Av ≤0,
ast−v < σ′(Av) (because t < Tv =v+σ′(Av)).
We are now able to establish the second part of Proposition 3.
Proof of (ii): We decompose Yt=
Z t 0
Wsds= Z t
0
Wsdτ(s) + Z t
0
Wsdτ′(s).
The change of variables s=Tv enables us to rewrite the first integral in the sum as Z t
0
Wsdτ(s) = Z τ(t)
0
WTvdv = Z τ(t)
0
(Bτ(Tv)+BT′v−τ(Tv))dv . Since τ(Tv) =v and Tv−τ(Tv) =Tv−v =σ′(Av), the right-hand side equals
Z τ(t) 0
(Bv+Av)dv = Z τ(t)
0
X˙vdv =Xτ(t),
where the first equality follows from (4).
Next, we write It′ := Rt
0 Wsdτ′(s), so that the process t → Yt−It′ = Xτ(t) is nonnegative.
Further, we know from Lemma 2(ii) thatt→It′is a non-increasing process and that the Stieltjes measure d(−It′) assigns no mass to Oc, and a fortiori is supported on the set of times t such thatYt−It′ =Xτ(t) = 0. An application of Skorohod’s reflection principle (see for instance [12]
on its page 239) enables to make the identification It′ = inf{Ys : 0≤ s≤t}=It. We conclude
that Xt =Xτ(Tt) =YTt −ITt.
Finally, we turn our attention to the third part of Proposition 3.
Proof of (iii): On the one hand, we have seen in the proof of part (ii) above that the infimum I of the free Langevin process Y can be expressed as
It= Z t
0
Wsdτ′(s) = Z t
0
1O(s)Wsds , where the second identity follows from Lemma 2.
On the other hand, we must have Wt ≤ 0 for every t such that Yt = It. Indeed, if we had Wt >0 for such a timet, then Y would be strictly increasing on some neighborhood oft, which is absurd. Now for every t≥0 such that Yt=It and Wt <0, Y is strictly decreasing on some interval [t, t′] with t′ > t, and thus Y =I on [t, t′]. Since the total time that W spends at 0 is zero, we also obtain
It = Z t
0
1{Ys=Is}Wsds .
Using again the fact that the total time thatW spends at 0 is zero, we deduce by comparison of these two expressions that with probability one, the random sets O and {s ≥0 : Ys = Is} coincide λ-almost everywhere. More precisely, recall the notation I :={s ≥ 0 : Ys = Is} and that the boundary ∂I = I\Io has zero Lebesgue measure. We now see that the open sets O and Io coincide a.s. As a consequence of Lemma 2,
τ(t) = Z t
0
1Oc(s)ds= Z t
0
1{Ys>Is}ds , t≥0,
and since T· is the right-continuous inverse of τ(·), this completes the proof.
4 Some comments and questions
We mentioned in the Introduction that second order differential equations with constraints of the type (1) and (2) may have multiple solutions even in the situation when the external force F is smooth. This was first pointed out by Bressan [3], who also made the conjecture that uniqueness holds when the force is a polynomial function of time. Schatzman [13] formulated the general setting of second order differential inclusions, and established a general theorem of existence of solutions. She also recovered independently Bressan’s example of a force of class C∞ for which such a system possesses multiple solutions. Percivale [10] was the first to show
that uniqueness holds for systems with only one degree of freedom, when the force is given by an analytic function of the time and depends neither on the position nor on the velocity of the particle, and then Schatzman [14] extended this to the much harder case when the force is an analytic function of time, position and velocity. Finally Ballard [1] considered more general discrete systems with several degrees of freedom and established that uniqueness always holds in the case when the force is analytic. We also refer to Maury [7, 8], Moreau [9] and Stewart [15] for numerical schemes for the computation of the motion of bodies systems with inelastic impacts.
For the convenience of the reader, we shall propose here simple example of an external force of classCk (for any fixed k ∈N) for which multiple solutions to (1) and (2) exist. Consider the increasing sequences 0<· · ·< sn< tn < sn+1 <· · · given by
sn:= 22n and tn:= 22n+1, n∈Z.
Then introduce the convex increasing functionα: [0,∞[→[0,∞[ which is linear on the intervals [sn, sn+1] and such that α(sn) = sk+3n . Similarly, let β : [0,∞[→ [0,∞[ denote the convex increasing function which is linear on the intervals [tn, tn+1] and such thatβ(tn) = tk+3n . Observe that α and β enjoy a property of self-similarity, namely
α(4u) = 4k+3α(u) and β(4u) = 4k+3β(u), u >0.
It should be obvious from a picture that there exists a function ϕ :]0,∞[→ R of class Ck+3, which is bounded from above by bothα and β, enjoys the same property of self-similarity, viz.
ϕ(4u) = 4k+3ϕ(u), and fulfills the following requirements :
ϕ(u) = α(u)⇔u=sn for somen ∈Z, ϕ(u) = β(u)⇔u=tn for somen ∈Z,
˙
ϕ(sn) = ˙α(sn+) for every n ∈Z,
˙
ϕ(tn) = ˙β(tn+) for every n∈Z.
More precisely, one constructs first a function ϕ which satisfies the preceding requirements on the interval [1,4], in such a way that for every ℓ= 0, . . . , k+ 3, theℓ-th derivative ϕ(ℓ) ofϕ has ϕ(ℓ)(4−) = 4k+3−ℓϕ(ℓ)(1+). Then ϕis extended to ]0,∞[ by self-similarity, and we setϕ(0) = 0.
Again by self-similarity, we get that ϕ is now of class Ck+2 on [0,∞[ with ϕ(ℓ)(0+) = 0 for every ℓ = 0, . . . , k+ 2. The requirements implies that Xu := α(u)−ϕ(u) solves (1) and (2) with Fu := −ϕ(u) and¨ Au := ˙α(u). Similarly, Xu := β(u)−ϕ(u) solves (1) and (2) with the same external force and Au := ˙β(u). Hence Equations (1) and (2) have at least two distinct solutions for Fu :=−ϕ(u).¨
Self-similarity is merely used above as a convenient tool for checking the regularity of the external force at 0, and a perusal of the argument reveals that a large class of counter-examples to uniqueness can be built by mimicking the preceding construction, using now an arbitrary strictly convex increasing function c : [0,∞[→ [0,∞[ (c(u) = uk+3 in the example above), and arbitrary increasing sequences (sn)n∈Z and (tn)n∈Z with no common point and such that limn→−∞sn = limn→−∞tn = 0. The external force F = −ϕ¨ may then no longer be smooth;
note that in any case, F has strong oscillations near zero, in the sense that F takes negative
and positive values at times arbitrarily close to 0. This may suggest that, informally, existence of multiple solutions to (1) and (2) could hold for quite general external forces F with strong oscillations. In this direction, recall that uniqueness of the solution has only been established for analytic external forces, see Ballard [1].
Theorem 2 is thus in sharp contrast with the preceding observations, even though the unique- ness is only stated there in a weak sense. Hence an important open question is to ask whether pathwise uniqueness holds for equations (1) and (4).
Another interesting problem in this vein is to decide whether or not the solution which has been constructed in Section 2 is adapted to the natural filtration of the Brownian motion (Bt)t≥0. One says that the solution isstrong in the case when the answer is positive. We refer to Tsirel’son [16] for a classical example of an SDE which has a unique weak solution, but no strong solution.
Acknowledgment : I would like to thank Patrick Ballard and Bertrand Maury for useful historical comments and references about the problem which motivated this work.
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